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Application of discrete wavelet transform to the solution of boundary value problems for quasi-linear parabolic equations. (English) Zbl 1286.65136

Summary: The wavelet method for solving the linear and quasi-linear parabolic equations under initial and boundary conditions is set out. By applying regular multi-resolution analysis and received formula for differentiating wavelet decompositions of functions of many variables the problem is reduced to a finite set of linear and accordingly nonlinear algebraic equations for the wavelet coefficients of the problem solution. The general scheme for finite-dimensional approximation in the regularization method is combined with the discrepancy principle. For quasi-linear parabolic equations the convergence rate of an approximate weak solution to a classical one is estimated.{ }The proposed method is used for constructing stable approximate wavelet decompositions of weak solutions to boundary value problems for the unsteady porous-medium flow equation with discontinuous coefficients and inexact data.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K59 Quasilinear parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data
76S05 Flows in porous media; filtration; seepage
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65T60 Numerical methods for wavelets

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